Zagier's Apéry Numbers as Critical L-Values of Weight-4 Modular Forms — E8 Intelligence Research

FINDING: Zagier's interpolated Apéry numbers for ζ(3) are critical L-values of a weight-4 modular form; the arXiv paper (1806.05207) extends this to all six sporadic sequences. MATH: - Apéry numbers: a(n) = Σₖ C(n,k)² C(n+k,k)², generating ζ(3) = (5/2) Σ (-1)^(n-1)/(n³ C(2n,n)) - Interpolation: A(n) → L(f, s) at critical points s ∈ {1, 2, 3} for weight-4 modular form f - Critical L-values: L(f, 1), L(f, 2), L(f, 3) for f ∈ S₄(Γ₀(N)) - Sporadic sequences (Zagier's six): Apéry (ζ(3)), and five others tied to weight-2/3/4 forms - Period relation: L(f, n) ~ Ω_f · (rational), Ω_f = real period of f CONNECTION: - Weight-4 modular forms ↔ period polynomials of degree 3; critical values sit at s = 1,2,3 — symmetric about s = 2 (center), reflecting functional equation s ↔ 4−s - Ratio L(f,1)/L(f,3) and L(f,2)/Ω_f encode arithmetic of the form; no direct 0.618/1.618 emergence in the cited result — but modular forms of weight k have Hodge structure with weights (k−1, 0) and (0, k−1), whose perio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com